This research proposal focuses on two areas of statistical methodology related to medical research, particularly cancer research. The first area is that of the design and analysis of Phase I clinical trials. Effort will be devoted to continued refinement of sampling algorithms and estimation and inference procedures developed under the previous proposal. In addition, research efforts will be devoted to two new topics. One of these is the use of multivariate models for toxicity parameters as a framework for estimation of the MTD. The toxicity parameters may represent continuous measurements, a discrete ordinal grading scale, or a combination of both. Various models will be assessed with respect to their computational feasibility, robustness, and the bias and efficiency of estimates of the MTD. Possible inference procedures will be considered. The impact of design on estimation and inference will be assessed. The other topic relates to design and analysis of Phase I trials in the multi- agent setting, where the goal is to estimate a range of combinations of drugs, or isotoxicity contour, producing a similar degree of toxicity. Various design strategies will be considered, as will analytic strategies. A major point of interest will be a comparison of contour estimates derived from modeling of the entire dose-response surface to those derived from interpolation between point estimates from one-dimensional dose-response models along the various design axes. Bias and efficiency of estimates of the isotoxicity contour will be evaluated, as well as sensitivity to modeling assumptions. Another area of research is that of diagnostic methodologies for the Cox proportional hazards regression model for censored survival data. Effort will be devoted to developing guidelines for identification of observations, or groups of observations, that have undue influence on the regression parameters. Techniques exist to measure the influence, but not necessarily to assess its degree of significance. Effort will also be devoted to the development of methods for determining the best use of the stratified proportional hazards model, when the stratification criteria are to be based on continuous covariates. Together such techniques will improve the ability of investigators to assess the adequacy and robustness of covariate models for survival time, such as those adjusting for known prognostic factors, or seeking to identify new ones.